Hitting time bounds for Brownian motion on a fractal

Krebs, William B.

doi:10.1090/S0002-9939-1993-1116263-X

Hitting time bounds for Brownian motion on a fractal
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by William B. Krebs

Proc. Amer. Math. Soc. 118 (1993), 223-232

DOI: https://doi.org/10.1090/S0002-9939-1993-1116263-X

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Abstract:

We calculate a bound on hitting times for Brownian motion defined on any nested fractal. We apply this bound to show that any such process is point recurrent. We then show that any diffusion on a nested fractal must have a transition density with respect to Hausdorff measure on the underlying fractal. We also prove that any Brownian motion on a nested fractal has a jointly continuous local time with a simple modulus of space-time continuity.

References

Meeting times for independent Markov chains

Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields 79 (1988), no. 4, 543–623. MR 966175, DOI 10.1007/BF00318785
R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR 0264757
R. K. Getoor and H. Kesten, Continuity of local times for Markov processes, Compositio Math. 24 (1972), 277–303. MR 310977

Random walks and diffusions defined on fractals

John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
William B. Krebs, A diffusion defined on a fractal state space, Stochastic Process. Appl. 37 (1991), no. 2, 199–212. MR 1102870, DOI 10.1016/0304-4149(91)90043-C
Shigeo Kusuoka, A diffusion process on a fractal, Probabilistic methods in mathematical physics (Katata/Kyoto, 1985) Academic Press, Boston, MA, 1987, pp. 251–274. MR 933827
Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 83 (1990), no. 420, iv+128. MR 988082, DOI 10.1090/memo/0420